A Mechanism for Anisotropic Reentry in Electrically Active Tissue

1992 was a good year for me. My wife and I, who had been married for seven years, had two young daughters, and we had just bought a house in Kensington, Maryland. While working at the National Institutes of Health I published eight papers in 1992, mostly about magnetic stimulation of nerves. My favorite paper from that year, however, was one about the heart: “A Mechanism for Anisotropic Reentry in Electrically Active Tissue” (Journal of Cardiovascular Electrophysiology, Volume 3, Pages 558–566). The lead author was Joshua Saypol, an engineering undergraduate at Brown University who would come home to Maryland each summer and work at NIH. Josh was a big, strong fellow, and handy to have around when we had heavy things to move. But he was also smart and hard-working, and we ended up publishing three papers together. The cardiac paper was the last of these, and the least cited (indeed, according to the Web of Science the paper has not been cited by anyone other than me for the last ten years). You can get the gist of it from the abstract.

Introduction: Numerical simulations of wavefront propagation were performed using a two-dimensional sheet of tissue with different anisotropy ratios in the intracellular and extracellular spaces.
Methods and Results: The tissue was represented by the bidomain model, and the active properties of the membrane were described by the Hodgkin-Huxley equations. Two successive stimuli, delivered through a single point electrode, resulted in the formation of a reentrant wavefront when the second stimulus was delivered during the vulnerable period of the first wavefront.
Conclusion: The mechanism for the development of reentry was that the bidomain tissue responded to point cathodal stimulation by depolarizing the tissue under the electrode in the direction perpendicular to the fiber axis, and hyperpolarizing the tissue in the direction parallel to the fiber axis. Such a distribution of depolarization and hyperpolarization modifies the refractory period of the action potential differently in each direction, resulting in block in the direction perpendicular to the fiber axis and leading to reentry and the formation of stable, rotating wavefronts.

The paper arises from two previous lines of research. First is the calculation of the transmembrane potential induced by a point stimulus, performed by Nestor Sepulveda, John Wikswo and me (“Current Injection Into a Two Dimensional Bidomain,” Biophysical Journal, Volume 55, Pages 987–999, 1989), which I discussed in a previous blog entry. We found that cardiac tissue is depolarized (positive transmembrane potential) under a cathode, but hyperpolarized (negative transmembrane potential) a millimeter or two from the cathode in each direction along the cardiac fibers (at what are nowadays called “virtual anodes”). That paper used a passive steady-state membrane, but in a subsequent paper I derived an algorithm to solve the bidomain equations including time dependence and an active model for the membrane kinetics (“Action Potential Propagation in a Thick Strand of Cardiac Muscle,” Circulation Research, Volume 68, Pages 162–173, 1991). Having this algorithm, I decided to investigate what effect the virtual anodes had on propagation following an extracellular stimulus. Both of these papers were based on the bidomain model, which is a mathematical model of the electrical properties of cardiac tissue that Russ Hobbie and I describe in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology.

Another reason I like the paper with Josh so much is that we collaborated with Arthur Winfree while writing it. I’ve written about our work with Winfree previously in this blog. Art had made a prediction about the induction of “reentry” (a cardiac arrhythmia) following a premature stimulus (a stimulus applied to tissue near the end of its refractory period). It provided the ideal hypothesis to test with our model. I remember vividly Josh coming to me with plots of the transmembrane potential showing the first signs of reentry, and how even after our discussions with Art we didn’t quite believe what we were seeing. Winfree helped Josh and I publish a preliminary letter about our calculation in the International Journal of Bifurcation and Chaos (Volume 1, Pages 927–928, 1991), which was just starting up. It has been almost ten years now since Art passed away, and I still miss him.

Astute readers (and I’m sure ALL readers of this blog are astute) will notice something odd in the abstract quoted above: “the active properties of the membrane were described by the Hodgkin-Huxley equations” (my italics). What are the Hodgkin-Huxley equations—which describe a squid nerve axon action potential (see Chapter 6 of Intermediate Physics for Medicine and Biology)—doing in a paper about cardiac tissue? This is a legitimate criticism, and one the reviewers raised, but surprisingly it didn’t prove fatal for publication of the article (although I was asked to change the title to the generic “...Electrically Active Tissue,” and you won’t find the word “cardiac” in the abstract). I used the Hodgkin-Huxley model because I didn’t know any better at the time. (In the paper, we claimed that “we used the Hodgkin-Huxley model instead of a myocardial membrane model because of a limitation of computer resources,” and that might be part of the reason too.) Models more appropriate for cardiac tissue (such as the Beeler-Reuter model that I used in later publications) were more complicated, and I wasn’t familiar with them. Besides, Josh and I were most interested in generic properties of reentry induction that would probably not be too sensitive to the membrane model (or so we told ourselves). I wonder now why we didn’t use the generic FitzHugh-Nagumo model, but we didn’t. Never again would reviewers let me get away with using the Hodgkin-Huxley model for cardiac tissue (and rightly so), and I suspect it is one of the reasons the paper is rarely cited anymore.

Nevertheless, the paper did make an important contribution to our understanding of the induction of reentry, which is why I like it so much. It was the first paper to present the idea of regions of hyperpolarization shortening the refractory period, and thereby creating regions of excitable tissue through which wave fronts can propagate. We state this clearly in the discussion.

The crucial point is that a premature stimulus causes an unusually-shaped transmembrane potential distribution that produces a directionally-dependent change of the refractory period, thereby creating a necessary condition for conduction block in one direction.

We go into more detail in the results

The depolarization wavefront is followed by a front of refractoriness. During the refractory period, the sodium channel inactivation gate (the h gate) opens slowly, while the potassium channel [activation gate] (the n gate) closes slowly; the tissue remains refractory until these two gates have recovered sufficiently. If a hyperpolarizing current is applied to the tissue during the refractory period, it will cause the h gate to open and the n gate to close more quickly than they normally would, thereby shortening the refractory period. Thus, when the tissue is stimulated, the refractory period is shortened in the area of hyperpolarization along the x axis [parallel to the fibers]. In the area along the y axis [perpendicular to the fibers] that is depolarized by the stimulus, on the other hand, the h and n gates move away from their resting values, and therefore the refractory period is lengthened. If the second stimulus is timed just right, it can take the tissue along the x direction out of the refractory period, while along the y direction the tissue remains unexcitable. Thus, the action potential elicited by the large depolarization directly below the electrode can propagate only in the x direction.

This idea influenced subsequent work by myself (“Nonsustained Reentry Following Successive Stimulation of Cardiac tissue Through a Unipolar Electrode,” Journal of Cardiovascular Electrophysiology, Volume 8, Pages 768–778, 1997) and others (Efimov et al., “Virtual Electrode-Induced Phase Singularity: A Basic Mechanism of Defibrillation Failure,” Circulation Research, Volume 82, Pages 918–925, 1998), and now, twenty years later, lies at the heart of the concept of “virtual electrodes” and their role during defibrillation (see, for instance: Efimov, Gray, and Roth, “Virtual Electrodes and Deexcitation: New Insights into Fibrillation Induction and Defibrillation,” Journal of Cardiovascular Electrophysiology, Volume 11, Pages 339–353, 2000). After I left NIH, Marc Lin, Wikswo and I confirmed experimentally this mechanism of reentry induction (“Quatrefoil Reentry in Myocardium: An Optical Imaging Study of the Induction Mechanisms,” Journal of Cardiovascular Electrophysiology, Volume 10, Pages 574–586, 1999).

The acknowledgments section of the paper brings back many memories.

Acknowledgments: We thank Art Winfree for his many ideas and suggestions, Peter Basser for his careful reading of the manuscript and Barry Bowman for his editorial assistance. The calculations were performed on the NIH Convex C240 computer. We thank the staff of the NIH computer center for their support.

First, of course, we mentioned Art’s contributions. Peter Basser, the inventor of MRI Diffusion Tensor Imaging, was a friend of mine at NIH, and we used to read each other’s papers before submission to a journal. Barry Bowman also worked at NIH. He was a former high school English teacher, and I’d always give him drafts of my papers for polishing. Much of what I know about writing English well I learned from him. I suspect my current laptop computer can calculate faster than the Convex C240 supercomputer could, but it was fairly powerful for the time. In 1992, Josh and I did our programming in FORTRAN. Some things never change.